In this paper we have derived the discreteness of measured
lengths in strong gravitational fields in a number of BH and
cosmological spacetimes.
In particular, we worked with the Schwarzschild and RN BH spacetimes as well as the FLRW cosmological spacetime.
For the BH spacetime, the strong gravity region is close to the singular region of the BH whereas for the FLRW spacetime it is in the inflationary epoch of the Universe. Remarkably, the derived discreteness turns out to be independent
of parameters such as the Schwarzschild mass, the RN charge, the KG mass or for that matter any particular epoch during the evolution of our Universe.
In addition, it agrees with the corresponding results derived for
zero gravity (flat spacetime) and weak gravity (weakly curved spacetime)
regimes and employing test fields of various spins
[1]}, [2]}, [3]}.
Furthermore, all length quantization conditions independently of the gravitational background are of the form
\(L \sim n\pi \alpha _{0}l_{pl}\)  modulo a natural number that depends on the specific gravitational background under study.
Finally, as in many previous works [4]}, [5]}, [6]}, [1]}, [8]}, [3]},
our results suggest the existence of a “new” length scale \(l_{new} \equiv \alpha _{0}l_{Pl}\) .
A recent theoretical work suggests \(\alpha _0 ={\cal O}(1)-{\cal O}(10)\)  [10]}. However, since no values of \(\alpha _0\)  are experimentally ruled out, we leave the possibility open for \(\alpha _0\)  to assume higher values, resulting in an intermediate length scale between the Planck and the electroweak length scales. This would lead to length quantization conditions of the form \(L\sim n\pi l_{new}\) .
This points towards the fundamental nature and in fact the
potential universality of the discreteness of measured spaces.
As we saw this follows directly from the application of GUP,
which is another
robust predictions of most candidate theories of QG.
As for the concrete computation to arrive at the above results,
we studied the massive KG equation in the background of a spacetime
with strong gravity.
Since QG effects cannot be ignored in this regime,
we included this by implementing GUP in the KG equation.
The form of GUP that we employed in our analysis is not manifestly Lorentz invariant. Therefore
one may ask as to why we did not try to implement a
relativistic version of GUP, e.g. as in Refs. [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}.
It should be stressed that while the aforesaid relativistic version of GUP is Lorentz covariant in
flat spacetimes, it still needs to be extended to curved spacetimes.
Furthermore, the form of GUP utilized here can be viewed as an “effective theory" from a fully covariant theory as described in Ref. [18]}.
Of course, our results may
undergo further modifications due to higher order QG effects
arising from the higher order terms in \(\alpha \)  which we have ignored. We hope to explore the implications of our results in the future.
